3.1789 \(\int \frac{(A+B x) \left (a^2+2 a b x+b^2 x^2\right )}{(d+e x)^{3/2}} \, dx\)
Optimal. Leaf size=124 \[ -\frac{2 b (d+e x)^{3/2} (-2 a B e-A b e+3 b B d)}{3 e^4}+\frac{2 \sqrt{d+e x} (b d-a e) (-a B e-2 A b e+3 b B d)}{e^4}+\frac{2 (b d-a e)^2 (B d-A e)}{e^4 \sqrt{d+e x}}+\frac{2 b^2 B (d+e x)^{5/2}}{5 e^4} \]
[Out]
(2*(b*d - a*e)^2*(B*d - A*e))/(e^4*Sqrt[d + e*x]) + (2*(b*d - a*e)*(3*b*B*d - 2*
A*b*e - a*B*e)*Sqrt[d + e*x])/e^4 - (2*b*(3*b*B*d - A*b*e - 2*a*B*e)*(d + e*x)^(
3/2))/(3*e^4) + (2*b^2*B*(d + e*x)^(5/2))/(5*e^4)
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Rubi [A] time = 0.157706, antiderivative size = 124, normalized size of antiderivative = 1.,
number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065
\[ -\frac{2 b (d+e x)^{3/2} (-2 a B e-A b e+3 b B d)}{3 e^4}+\frac{2 \sqrt{d+e x} (b d-a e) (-a B e-2 A b e+3 b B d)}{e^4}+\frac{2 (b d-a e)^2 (B d-A e)}{e^4 \sqrt{d+e x}}+\frac{2 b^2 B (d+e x)^{5/2}}{5 e^4} \]
Antiderivative was successfully verified.
[In] Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2))/(d + e*x)^(3/2),x]
[Out]
(2*(b*d - a*e)^2*(B*d - A*e))/(e^4*Sqrt[d + e*x]) + (2*(b*d - a*e)*(3*b*B*d - 2*
A*b*e - a*B*e)*Sqrt[d + e*x])/e^4 - (2*b*(3*b*B*d - A*b*e - 2*a*B*e)*(d + e*x)^(
3/2))/(3*e^4) + (2*b^2*B*(d + e*x)^(5/2))/(5*e^4)
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Rubi in Sympy [A] time = 53.5373, size = 122, normalized size = 0.98 \[ \frac{2 B b^{2} \left (d + e x\right )^{\frac{5}{2}}}{5 e^{4}} + \frac{2 b \left (d + e x\right )^{\frac{3}{2}} \left (A b e + 2 B a e - 3 B b d\right )}{3 e^{4}} + \frac{2 \sqrt{d + e x} \left (a e - b d\right ) \left (2 A b e + B a e - 3 B b d\right )}{e^{4}} - \frac{2 \left (A e - B d\right ) \left (a e - b d\right )^{2}}{e^{4} \sqrt{d + e x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)*(b**2*x**2+2*a*b*x+a**2)/(e*x+d)**(3/2),x)
[Out]
2*B*b**2*(d + e*x)**(5/2)/(5*e**4) + 2*b*(d + e*x)**(3/2)*(A*b*e + 2*B*a*e - 3*B
*b*d)/(3*e**4) + 2*sqrt(d + e*x)*(a*e - b*d)*(2*A*b*e + B*a*e - 3*B*b*d)/e**4 -
2*(A*e - B*d)*(a*e - b*d)**2/(e**4*sqrt(d + e*x))
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Mathematica [A] time = 0.167872, size = 135, normalized size = 1.09 \[ \frac{30 a^2 e^2 (-A e+2 B d+B e x)+20 a b e \left (3 A e (2 d+e x)+B \left (-8 d^2-4 d e x+e^2 x^2\right )\right )+2 b^2 \left (5 A e \left (-8 d^2-4 d e x+e^2 x^2\right )+3 B \left (16 d^3+8 d^2 e x-2 d e^2 x^2+e^3 x^3\right )\right )}{15 e^4 \sqrt{d+e x}} \]
Antiderivative was successfully verified.
[In] Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2))/(d + e*x)^(3/2),x]
[Out]
(30*a^2*e^2*(2*B*d - A*e + B*e*x) + 20*a*b*e*(3*A*e*(2*d + e*x) + B*(-8*d^2 - 4*
d*e*x + e^2*x^2)) + 2*b^2*(5*A*e*(-8*d^2 - 4*d*e*x + e^2*x^2) + 3*B*(16*d^3 + 8*
d^2*e*x - 2*d*e^2*x^2 + e^3*x^3)))/(15*e^4*Sqrt[d + e*x])
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Maple [A] time = 0.013, size = 169, normalized size = 1.4 \[ -{\frac{-6\,B{x}^{3}{b}^{2}{e}^{3}-10\,A{b}^{2}{e}^{3}{x}^{2}-20\,Bab{e}^{3}{x}^{2}+12\,B{b}^{2}d{e}^{2}{x}^{2}-60\,Axab{e}^{3}+40\,Ax{b}^{2}d{e}^{2}-30\,Bx{a}^{2}{e}^{3}+80\,Bxabd{e}^{2}-48\,B{b}^{2}{d}^{2}ex+30\,A{a}^{2}{e}^{3}-120\,Aabd{e}^{2}+80\,A{b}^{2}{d}^{2}e-60\,Bd{e}^{2}{a}^{2}+160\,B{d}^{2}abe-96\,B{b}^{2}{d}^{3}}{15\,{e}^{4}}{\frac{1}{\sqrt{ex+d}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)*(b^2*x^2+2*a*b*x+a^2)/(e*x+d)^(3/2),x)
[Out]
-2/15*(-3*B*b^2*e^3*x^3-5*A*b^2*e^3*x^2-10*B*a*b*e^3*x^2+6*B*b^2*d*e^2*x^2-30*A*
a*b*e^3*x+20*A*b^2*d*e^2*x-15*B*a^2*e^3*x+40*B*a*b*d*e^2*x-24*B*b^2*d^2*e*x+15*A
*a^2*e^3-60*A*a*b*d*e^2+40*A*b^2*d^2*e-30*B*a^2*d*e^2+80*B*a*b*d^2*e-48*B*b^2*d^
3)/(e*x+d)^(1/2)/e^4
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Maxima [A] time = 0.724418, size = 225, normalized size = 1.81 \[ \frac{2 \,{\left (\frac{3 \,{\left (e x + d\right )}^{\frac{5}{2}} B b^{2} - 5 \,{\left (3 \, B b^{2} d -{\left (2 \, B a b + A b^{2}\right )} e\right )}{\left (e x + d\right )}^{\frac{3}{2}} + 15 \,{\left (3 \, B b^{2} d^{2} - 2 \,{\left (2 \, B a b + A b^{2}\right )} d e +{\left (B a^{2} + 2 \, A a b\right )} e^{2}\right )} \sqrt{e x + d}}{e^{3}} + \frac{15 \,{\left (B b^{2} d^{3} - A a^{2} e^{3} -{\left (2 \, B a b + A b^{2}\right )} d^{2} e +{\left (B a^{2} + 2 \, A a b\right )} d e^{2}\right )}}{\sqrt{e x + d} e^{3}}\right )}}{15 \, e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)*(B*x + A)/(e*x + d)^(3/2),x, algorithm="maxima")
[Out]
2/15*((3*(e*x + d)^(5/2)*B*b^2 - 5*(3*B*b^2*d - (2*B*a*b + A*b^2)*e)*(e*x + d)^(
3/2) + 15*(3*B*b^2*d^2 - 2*(2*B*a*b + A*b^2)*d*e + (B*a^2 + 2*A*a*b)*e^2)*sqrt(e
*x + d))/e^3 + 15*(B*b^2*d^3 - A*a^2*e^3 - (2*B*a*b + A*b^2)*d^2*e + (B*a^2 + 2*
A*a*b)*d*e^2)/(sqrt(e*x + d)*e^3))/e
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Fricas [A] time = 0.291765, size = 209, normalized size = 1.69 \[ \frac{2 \,{\left (3 \, B b^{2} e^{3} x^{3} + 48 \, B b^{2} d^{3} - 15 \, A a^{2} e^{3} - 40 \,{\left (2 \, B a b + A b^{2}\right )} d^{2} e + 30 \,{\left (B a^{2} + 2 \, A a b\right )} d e^{2} -{\left (6 \, B b^{2} d e^{2} - 5 \,{\left (2 \, B a b + A b^{2}\right )} e^{3}\right )} x^{2} +{\left (24 \, B b^{2} d^{2} e - 20 \,{\left (2 \, B a b + A b^{2}\right )} d e^{2} + 15 \,{\left (B a^{2} + 2 \, A a b\right )} e^{3}\right )} x\right )}}{15 \, \sqrt{e x + d} e^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)*(B*x + A)/(e*x + d)^(3/2),x, algorithm="fricas")
[Out]
2/15*(3*B*b^2*e^3*x^3 + 48*B*b^2*d^3 - 15*A*a^2*e^3 - 40*(2*B*a*b + A*b^2)*d^2*e
+ 30*(B*a^2 + 2*A*a*b)*d*e^2 - (6*B*b^2*d*e^2 - 5*(2*B*a*b + A*b^2)*e^3)*x^2 +
(24*B*b^2*d^2*e - 20*(2*B*a*b + A*b^2)*d*e^2 + 15*(B*a^2 + 2*A*a*b)*e^3)*x)/(sqr
t(e*x + d)*e^4)
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Sympy [A] time = 27.6016, size = 2734, normalized size = 22.05 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)*(b**2*x**2+2*a*b*x+a**2)/(e*x+d)**(3/2),x)
[Out]
-2*A*a**2/(e*sqrt(d + e*x)) + 2*A*a*b*Piecewise((4*d/(e**2*sqrt(d + e*x)) + 2*x/
(e*sqrt(d + e*x)), Ne(e, 0)), (x**2/(2*d**(3/2)), True)) + A*b**2*(-16*d**(19/2)
*sqrt(1 + e*x/d)/(3*d**8*e**3 + 9*d**7*e**4*x + 9*d**6*e**5*x**2 + 3*d**5*e**6*x
**3) + 16*d**(19/2)/(3*d**8*e**3 + 9*d**7*e**4*x + 9*d**6*e**5*x**2 + 3*d**5*e**
6*x**3) - 40*d**(17/2)*e*x*sqrt(1 + e*x/d)/(3*d**8*e**3 + 9*d**7*e**4*x + 9*d**6
*e**5*x**2 + 3*d**5*e**6*x**3) + 48*d**(17/2)*e*x/(3*d**8*e**3 + 9*d**7*e**4*x +
9*d**6*e**5*x**2 + 3*d**5*e**6*x**3) - 30*d**(15/2)*e**2*x**2*sqrt(1 + e*x/d)/(
3*d**8*e**3 + 9*d**7*e**4*x + 9*d**6*e**5*x**2 + 3*d**5*e**6*x**3) + 48*d**(15/2
)*e**2*x**2/(3*d**8*e**3 + 9*d**7*e**4*x + 9*d**6*e**5*x**2 + 3*d**5*e**6*x**3)
- 4*d**(13/2)*e**3*x**3*sqrt(1 + e*x/d)/(3*d**8*e**3 + 9*d**7*e**4*x + 9*d**6*e*
*5*x**2 + 3*d**5*e**6*x**3) + 16*d**(13/2)*e**3*x**3/(3*d**8*e**3 + 9*d**7*e**4*
x + 9*d**6*e**5*x**2 + 3*d**5*e**6*x**3) + 2*d**(11/2)*e**4*x**4*sqrt(1 + e*x/d)
/(3*d**8*e**3 + 9*d**7*e**4*x + 9*d**6*e**5*x**2 + 3*d**5*e**6*x**3)) + B*a**2*P
iecewise((4*d/(e**2*sqrt(d + e*x)) + 2*x/(e*sqrt(d + e*x)), Ne(e, 0)), (x**2/(2*
d**(3/2)), True)) + 2*B*a*b*(-16*d**(19/2)*sqrt(1 + e*x/d)/(3*d**8*e**3 + 9*d**7
*e**4*x + 9*d**6*e**5*x**2 + 3*d**5*e**6*x**3) + 16*d**(19/2)/(3*d**8*e**3 + 9*d
**7*e**4*x + 9*d**6*e**5*x**2 + 3*d**5*e**6*x**3) - 40*d**(17/2)*e*x*sqrt(1 + e*
x/d)/(3*d**8*e**3 + 9*d**7*e**4*x + 9*d**6*e**5*x**2 + 3*d**5*e**6*x**3) + 48*d*
*(17/2)*e*x/(3*d**8*e**3 + 9*d**7*e**4*x + 9*d**6*e**5*x**2 + 3*d**5*e**6*x**3)
- 30*d**(15/2)*e**2*x**2*sqrt(1 + e*x/d)/(3*d**8*e**3 + 9*d**7*e**4*x + 9*d**6*e
**5*x**2 + 3*d**5*e**6*x**3) + 48*d**(15/2)*e**2*x**2/(3*d**8*e**3 + 9*d**7*e**4
*x + 9*d**6*e**5*x**2 + 3*d**5*e**6*x**3) - 4*d**(13/2)*e**3*x**3*sqrt(1 + e*x/d
)/(3*d**8*e**3 + 9*d**7*e**4*x + 9*d**6*e**5*x**2 + 3*d**5*e**6*x**3) + 16*d**(1
3/2)*e**3*x**3/(3*d**8*e**3 + 9*d**7*e**4*x + 9*d**6*e**5*x**2 + 3*d**5*e**6*x**
3) + 2*d**(11/2)*e**4*x**4*sqrt(1 + e*x/d)/(3*d**8*e**3 + 9*d**7*e**4*x + 9*d**6
*e**5*x**2 + 3*d**5*e**6*x**3)) + B*b**2*(32*d**(45/2)*sqrt(1 + e*x/d)/(5*d**20*
e**4 + 30*d**19*e**5*x + 75*d**18*e**6*x**2 + 100*d**17*e**7*x**3 + 75*d**16*e**
8*x**4 + 30*d**15*e**9*x**5 + 5*d**14*e**10*x**6) - 32*d**(45/2)/(5*d**20*e**4 +
30*d**19*e**5*x + 75*d**18*e**6*x**2 + 100*d**17*e**7*x**3 + 75*d**16*e**8*x**4
+ 30*d**15*e**9*x**5 + 5*d**14*e**10*x**6) + 176*d**(43/2)*e*x*sqrt(1 + e*x/d)/
(5*d**20*e**4 + 30*d**19*e**5*x + 75*d**18*e**6*x**2 + 100*d**17*e**7*x**3 + 75*
d**16*e**8*x**4 + 30*d**15*e**9*x**5 + 5*d**14*e**10*x**6) - 192*d**(43/2)*e*x/(
5*d**20*e**4 + 30*d**19*e**5*x + 75*d**18*e**6*x**2 + 100*d**17*e**7*x**3 + 75*d
**16*e**8*x**4 + 30*d**15*e**9*x**5 + 5*d**14*e**10*x**6) + 396*d**(41/2)*e**2*x
**2*sqrt(1 + e*x/d)/(5*d**20*e**4 + 30*d**19*e**5*x + 75*d**18*e**6*x**2 + 100*d
**17*e**7*x**3 + 75*d**16*e**8*x**4 + 30*d**15*e**9*x**5 + 5*d**14*e**10*x**6) -
480*d**(41/2)*e**2*x**2/(5*d**20*e**4 + 30*d**19*e**5*x + 75*d**18*e**6*x**2 +
100*d**17*e**7*x**3 + 75*d**16*e**8*x**4 + 30*d**15*e**9*x**5 + 5*d**14*e**10*x*
*6) + 462*d**(39/2)*e**3*x**3*sqrt(1 + e*x/d)/(5*d**20*e**4 + 30*d**19*e**5*x +
75*d**18*e**6*x**2 + 100*d**17*e**7*x**3 + 75*d**16*e**8*x**4 + 30*d**15*e**9*x*
*5 + 5*d**14*e**10*x**6) - 640*d**(39/2)*e**3*x**3/(5*d**20*e**4 + 30*d**19*e**5
*x + 75*d**18*e**6*x**2 + 100*d**17*e**7*x**3 + 75*d**16*e**8*x**4 + 30*d**15*e*
*9*x**5 + 5*d**14*e**10*x**6) + 290*d**(37/2)*e**4*x**4*sqrt(1 + e*x/d)/(5*d**20
*e**4 + 30*d**19*e**5*x + 75*d**18*e**6*x**2 + 100*d**17*e**7*x**3 + 75*d**16*e*
*8*x**4 + 30*d**15*e**9*x**5 + 5*d**14*e**10*x**6) - 480*d**(37/2)*e**4*x**4/(5*
d**20*e**4 + 30*d**19*e**5*x + 75*d**18*e**6*x**2 + 100*d**17*e**7*x**3 + 75*d**
16*e**8*x**4 + 30*d**15*e**9*x**5 + 5*d**14*e**10*x**6) + 92*d**(35/2)*e**5*x**5
*sqrt(1 + e*x/d)/(5*d**20*e**4 + 30*d**19*e**5*x + 75*d**18*e**6*x**2 + 100*d**1
7*e**7*x**3 + 75*d**16*e**8*x**4 + 30*d**15*e**9*x**5 + 5*d**14*e**10*x**6) - 19
2*d**(35/2)*e**5*x**5/(5*d**20*e**4 + 30*d**19*e**5*x + 75*d**18*e**6*x**2 + 100
*d**17*e**7*x**3 + 75*d**16*e**8*x**4 + 30*d**15*e**9*x**5 + 5*d**14*e**10*x**6)
+ 16*d**(33/2)*e**6*x**6*sqrt(1 + e*x/d)/(5*d**20*e**4 + 30*d**19*e**5*x + 75*d
**18*e**6*x**2 + 100*d**17*e**7*x**3 + 75*d**16*e**8*x**4 + 30*d**15*e**9*x**5 +
5*d**14*e**10*x**6) - 32*d**(33/2)*e**6*x**6/(5*d**20*e**4 + 30*d**19*e**5*x +
75*d**18*e**6*x**2 + 100*d**17*e**7*x**3 + 75*d**16*e**8*x**4 + 30*d**15*e**9*x*
*5 + 5*d**14*e**10*x**6) + 6*d**(31/2)*e**7*x**7*sqrt(1 + e*x/d)/(5*d**20*e**4 +
30*d**19*e**5*x + 75*d**18*e**6*x**2 + 100*d**17*e**7*x**3 + 75*d**16*e**8*x**4
+ 30*d**15*e**9*x**5 + 5*d**14*e**10*x**6) + 2*d**(29/2)*e**8*x**8*sqrt(1 + e*x
/d)/(5*d**20*e**4 + 30*d**19*e**5*x + 75*d**18*e**6*x**2 + 100*d**17*e**7*x**3 +
75*d**16*e**8*x**4 + 30*d**15*e**9*x**5 + 5*d**14*e**10*x**6))
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GIAC/XCAS [A] time = 0.282439, size = 296, normalized size = 2.39 \[ \frac{2}{15} \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} B b^{2} e^{16} - 15 \,{\left (x e + d\right )}^{\frac{3}{2}} B b^{2} d e^{16} + 45 \, \sqrt{x e + d} B b^{2} d^{2} e^{16} + 10 \,{\left (x e + d\right )}^{\frac{3}{2}} B a b e^{17} + 5 \,{\left (x e + d\right )}^{\frac{3}{2}} A b^{2} e^{17} - 60 \, \sqrt{x e + d} B a b d e^{17} - 30 \, \sqrt{x e + d} A b^{2} d e^{17} + 15 \, \sqrt{x e + d} B a^{2} e^{18} + 30 \, \sqrt{x e + d} A a b e^{18}\right )} e^{\left (-20\right )} + \frac{2 \,{\left (B b^{2} d^{3} - 2 \, B a b d^{2} e - A b^{2} d^{2} e + B a^{2} d e^{2} + 2 \, A a b d e^{2} - A a^{2} e^{3}\right )} e^{\left (-4\right )}}{\sqrt{x e + d}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)*(B*x + A)/(e*x + d)^(3/2),x, algorithm="giac")
[Out]
2/15*(3*(x*e + d)^(5/2)*B*b^2*e^16 - 15*(x*e + d)^(3/2)*B*b^2*d*e^16 + 45*sqrt(x
*e + d)*B*b^2*d^2*e^16 + 10*(x*e + d)^(3/2)*B*a*b*e^17 + 5*(x*e + d)^(3/2)*A*b^2
*e^17 - 60*sqrt(x*e + d)*B*a*b*d*e^17 - 30*sqrt(x*e + d)*A*b^2*d*e^17 + 15*sqrt(
x*e + d)*B*a^2*e^18 + 30*sqrt(x*e + d)*A*a*b*e^18)*e^(-20) + 2*(B*b^2*d^3 - 2*B*
a*b*d^2*e - A*b^2*d^2*e + B*a^2*d*e^2 + 2*A*a*b*d*e^2 - A*a^2*e^3)*e^(-4)/sqrt(x
*e + d)